A flat head drill bit is used to drill holes of small depth. The drill bit can drill holes with a diameter size of (40 ± 0.5)mm. The depth of the hole has an uncertainty of ±1mm.

Question

A flat head drill bit is used to drill holes of small depth. The drill bit can drill holes with a diameter size of (40 ± 0.5)mm. The depth of the hole has an uncertainty of ±1mm.

If the total percentage uncertainty of the volume made by the hole is less than 10%, what is the minimum depth, in mm, which the drill bit can make?

My working:
V=π(r^2)d => d= 4V/(πD^2)
[V=Volume; r=radius; d=depth; D=Diameter]
△d/d = △v/v + 2(△D/D)
1/d = 10/100 + 2(0.5/40)
d=8

But the options given are 15, 14, 13 and 12. What am I doing wrong?

Answer 1

There's a small mistake in your calculation. Let me correct it for you.

The total percentage uncertainty formula for volume is given by:

ΔV/V = 2(ΔD/D) + Δd/d

Given that the total percentage uncertainty is less than 10%, we can rewrite this equation as:

2(ΔD/D) + Δd/d < 10/100

Substituting the values:

2(0.5/40) + Δd/(40 ± 0.5) < 0.1

0.025 + Δd/(40 ± 0.5) < 0.1

Now, to find the minimum depth, we would consider the worst case scenario, which is when the diameter is at its lowest value (40 - 0.5) and the depth is at its highest value (let's call it d_max).

0.025 + d_max/(40 - 0.5) < 0.1

0.025 + d_max/39.5 < 0.1

0.025*39.5 + d_max < 0.1*39.5

d_max < 0.1*39.5 - 0.025*39.5

d_max < 3.95 - 0.9875

d_max < 2.9625

Therefore, the minimum depth, rounded to the nearest whole number, is 2.

So, the correct answer is 2 mm. None of the options provided match this result. It seems like there may be an error either in the options or in the given information.

Answer 2

Your working is correct so far. You calculated the percentage uncertainty using the formula:

△d/d = △v/v + 2(△D/D)

Where △d is the uncertainty in the depth, △v is the uncertainty in the volume, and △D is the uncertainty in the diameter.

You correctly substituted the values into the formula:

1/d = 10/100 + 2(0.5/40)

Now, let's simplify the equation further:

1/d = 0.1 + 0.025

1/d = 0.125

To find d, we can take the reciprocal of both sides of the equation:

d = 1/0.125

d = 8

According to your calculations, the minimum depth should be 8 mm. However, the given options are different. This suggests that there might be an error in either the answer choices or the information provided.

I would recommend double-checking the given options or reviewing the problem statement to see if any information was missed or if there is a mistake in the options.